Show that for any triangle it is always possible to construct 3
touching circles with centres at the vertices. Is it possible to
construct touching circles centred at the vertices of any polygon?

An Introduction to Magic Squares

Stage: 2, 3 and 4

Article by NRICH team

Published February 2000,July 2007,August 2007,February 2011.

Magic Squares are square grids with a special arrangement of
numbers in them. These numbers are special because every row,
column and diagonal adds up to the same number. So for the example
below, 15 is the magic number. Could you work this out just from
knowing that the square uses the numbers from 1 to 9?

Also, the two numbers that are opposite each other across the
centre number will add up to the same number. So in the square
above, 8 + 2 = 10 , 6 + 4 = 10, 1 + 9 = 10 and 3 + 7 = 10. Why is
this?

The "order" of a magic square tells how many rows or columns it
has. So a square with 3 rows and columns is Order 3, and a square
with 4 rows and columns is Order 4 and so on. If you'd like to find
out more about how to make up your own magic squares, and the
mathematics behind it all, you can go to some other pages on the
website such as
Magic Squares and
Magic Squares II .

So the numbers in the Magic Square are special, but why are they
called magic? It seems that from ancient times they were connected
with the supernatural and magical world. The earliest record of
magic squares is from China in about 2200 BC. and is called
"Lo-Shu". There's a legend that says that the Emperor Yu saw this
magic square on the back of a divine tortoise in the Yellow River.

The black knots show even numbers and the white knots show odd
numbers. Look closely and you'll see that this ancient magic square
is the same as our example above. Magic squares were first
mentioned in the Western world in the work of Theon of Smyrna. They
were also used by Arab astrologers in the 9th century to help work
out horoscopes. The work of the Greek mathematician Moschopoulos in
1300 A.D. help to spread knowledge about magic squares. So here we
are now, more than 700 years later, and teachers are using
them in class for problem solving and practising
addition.

You can make similar magic squares, of order 3, using different
numbers. Can you see any patterns in the numbers that work?

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities
can be found here.